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The From omega to epsilon nought blog lists out all the ordinals between $$\omega$$ to $$\epsilon_0$$ in detail assuming n=3 when substituting finite values for $$\omega$$. This blog summarises the detail and calculates the number of steps in the progression.

The progression is used to compare big numbers in the Strong D Function blog post.

## Recap of all steps from omega to epsilon nought

1. add $$\omega$$ 9 times to get $$\omega^{\omega}$$

2. add $$\omega^{\omega}$$ 3 times to get $$\omega^{\omega+1}$$

3. add $$\omega^{\omega+1}$$ 3 times to get $$\omega^{\omega+2}$$

4. add $$\omega^{\omega+2}$$ 3 times to get $$\omega^{\omega.2}$$

5. add $$\omega^{\omega.2}$$ 3 times to get $$\omega^{\omega.2+1}$$

6. add $$\omega^{\omega.2+1}$$ 3 times to get $$\omega^{\omega.2+2}$$

7. add $$\omega^{\omega.2+2}$$ 3 times to get $$\omega^{\omega^2}$$

8. add $$\omega^{\omega^2}$$ 3 times to get $$\omega^{\omega^2+1}$$

9. add $$\omega^{\omega^2+1}$$ 3 times to get $$\omega^{\omega^2+2}$$

10. add $$\omega^{\omega^2+2}$$ 3 times to get $$\omega^{\omega^2+\omega}$$

11. add $$\omega^{\omega^2+\omega}$$ 3 times to get $$\omega^{\omega^2+\omega+1}$$

12. add $$\omega^{\omega^2+\omega+1}$$ 3 times to get $$\omega^{\omega^2+\omega+2}$$

13. add $$\omega^{\omega^2+\omega+2}$$ 3 times to get $$\omega^{\omega^2+\omega.2}$$

14. add $$\omega^{\omega^2+\omega.2}$$ 3 times to get $$\omega^{\omega^2+\omega.2+1}$$

15. add $$\omega^{\omega^2+\omega.2+1}$$ 3 times to get $$\omega^{\omega^2+\omega.2+2}$$

16. add $$\omega^{\omega^2+\omega.2+2}$$ 3 times to get $$\omega^{\omega^2.\omega.2+\omega}$$

17. add $$\omega^{\omega^2.2}$$ 3 times to get $$\omega^{\omega^2.2+1}$$

18. add $$\omega^{\omega^2.2+1}$$ 3 times to get $$\omega^{\omega^2.2+2}$$

19. add $$\omega^{\omega^2.2+2}$$ 3 times to get $$\omega^{\omega^2.2+\omega}$$

20. add $$\omega^{\omega^2.2+\omega}$$ 3 times to get $$\omega^{\omega^2.2+\omega+1}$$

21. add $$\omega^{\omega^2.2+\omega+1}$$ 3 times to get $$\omega^{\omega^2.2+\omega+2}$$

22. add $$\omega^{\omega^2.2+\omega+2}$$ 3 times to get $$\omega^{\omega^2.2+\omega.2}$$

23. add $$\omega^{\omega^2.2+\omega.2}$$ 3 times to get $$\omega^{\omega^2.2+\omega.2+1}$$

24. add $$\omega^{\omega^2.2+\omega.2+1}$$ 3 times to get $$\omega^{\omega^2.2+\omega.2+2}$$

25. add $$\omega^{\omega^2.2+\omega.2+2}$$ 3 times to get $$\epsilon_0$$

## Calculating the total number of steps

All of the above steps are recursive. For example, Step 8 requires adding $$\omega^{\omega^2}$$ 3 times. But adding $$\omega^{\omega^2}$$ once requires all of Step 7 (adding $$\omega^{\omega.2+2}$$ 3 times) and so on. This continues all the way to Step 1 and adding $$\omega$$ 9 times.

Therefore the number of steps in total is $$3^24.9 = 3^{26} = 3^{3^3-1}$$

The general rule for any n other than 3 is $$n^{n\uparrow\uparrow(n-1)-1}$$